Representations and irreducible subrepresentations

[glmuelle]

I don't know how to do the following homework:

Let Gbe a finite group and let \rho : G \rightarrow GL(E)be a finite-dimensional
faithful complex representation, i.e. ker \rho = 1. For any irreducible complex representation \piof G, show that there exists k \geq 1 such that \pi is an irreducible subrepresentation of \rho_k = \rho \otimes \dots \otimes \rho (k times).

Hint: Let a_k = \langle \chi_{\rho_k}, \chi_\pi \rangle be the multiplicity of \pi in \rho_k; compute the power series f(X) = \sum_{k \geq 0} a_k X^k and show that it is non-zero.


The definition of irreducible representation is that \rho: G \righarrow GL(E) is irreducible if it has no subrepresentation which means that E has no proper subspace \neq 0 that is stable under \rho.

\otimes is the tensor product.

\chi_\pi is the character of \pi which is the trace of \pi.


I have no attempt at a solution because I don't even know where to start. For example, can someone tell me what the "multiplicity of \pi in \rho_k is? And what are the angle brackets? First I thought it's an inner product but since characters are scalars that doesn't make any sense. Many thanks for your help, I really appreciate it!

 

[Stephen Tashi]

I fixed the latex tags for you. Let's hope a group representation expert shows up soon.

(In case you didn't know this. The tag for LaTex on this forum is the "tex" inside square braces at the beginning and "/tex" inside square braces at the end. There is a bug with the way LaTex is cached. When you edit a message, you must use "preview post" and then, after the page appears, you must reload it with your browsers refresh button. If you edit a post you must do a similar process. Otherwise the LaTex looks screwy.)

I don't know how to do the following homework:

Let G be a finite group and let \rho : G \rightarrow GL(E) be a finite-dimensional
faithful complex representation, i.e. ker \rho = 1. For any irreducible complex representation \pi of G, show that there exists k \geq 1 such that \pi is an irreducible subrepresentation of \rho_k = \rho \otimes \dots \otimes \rho (k times).

Hint: Let a_k = \langle \chi_{\rho_k}, \chi_\pi \rangle be the multiplicity of \pi in \rho_k; compute the power series f(X) = \sum_{k \geq 0} a_k X^k and show that it is non-zero.


The definition of irreducible representation is that \rho: G \rightarrow GL(E) is irreducible if it has no subrepresentation which means that E has no proper subspace \neq 0 that is stable under \rho.

\otimes is the tensor product.

\chi_\pi is the character of \pi which is the trace of \pi.


I have no attempt at a solution because I don't even know where to start. For example, can someone tell me what the "multiplicity of \pi in \rho_k is? And what are the angle brackets? First I thought it's an inner product but since characters are scalars that doesn't make any sense. Many thanks for your help, I really appreciate it!

 

[glmuelle]

Thank you Stephen. I had tried to preview the post before I posted but somehow I got 'unknown error' three times. Then I just submitted it without previewing, I'm sorry for this.